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Sadhana Vol. 28, Parts 3 & 4,June/August 2003, pp. 763–782. © Printed in India

Convergent beam electron diffraction – A novel techniquefor materials characterisation at sub-microscopic levels

M VIJAYALAKSHMI, S SAROJA and R MYTHILI

Physical Metallurgy Section, Materials Characterisation Group, Indira GandhiCentre for Atomic Research, Kalpakkam 603 102, Indiae-mail: [email protected]

Abstract. This paper presents a review of the developments in ConvergentBeam Electron Diffraction (CBED), a technique widely used for determination ofstructure, symmetry details and atom positions in a crystal as small as 20Å in size.The understanding of this technique is related to the rapid advancements in the fieldof transmission electron microscopy with respect to development of coherent, finerprobes and electron optics for higher spatial resolution. Energy filtering devicesenable imaging of several finer features in the CBED pattern from which usefulinformation about a crystal can be obtained. These include (i) three-dimensionalinformation about the reciprocal lattice, (ii) point and space group symmetry details,(iii) lattice parameter from regions as fine as 2nm, (iv) atom positions within a unitcell and (v) defects in crystals and (vi) thickness. Due to abundant data obtainedfrom microscopic regions, this technique is unique and finds wide application inmaterials characterization. It has been used for studying problems like identificationof the presence of lattice strain, identification of point defects etc. in a material usedoften in the nuclear industry, namely 9Cr–1Mo steel. The present paper providesthe current status of CBED starting from its historical development, the informationthat can be obtained and its use in a variety of applications.

Keywords. Convergent beam electron diffraction; nanodiffraction; holography;reciprocal lattice, lattice parameter; convergence angle.

1. Introduction

Convergent beam electron diffraction, generally referred to as CBED, is one of the mostpowerful techniques for the determination of crystal structure in the field of transmissionelectron microscopy (TEM). This technique is used for fingerprinting crystals as fine as 20Å,by determining a number of parameters ranging from the symmetry of the crystal to theposition of atoms within the unit cell of the diffracting crystal. The technique of CBED wasdiscovered in 1939 by Kossel & Mollenstedt (1939), who obtained remarkably good patterns,especially considering that they had to work with large probes, with small convergence angles.The subsequent development of STEM units made it possible to obtain finer probes withlarger angles of convergence, making CBED more popular. Concurrently, development ofcoherent, powerful electron sources like the LaB6, field emission and field ion guns (Crewe &

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Eggenberger 1968), improved design of etc. lens systems with less aberration, imaging usingenergy filtering devices (Lanio 1986) etc. have made this technique very powerful. Parallelconcerted efforts were focused on the understanding of the principle of imaging, the originof fine, additional features that appear in CBED patterns and methods to extract additionaldetails about the crystal using this information. In fact, the major and long standing drawbackof electron diffraction, in comparison with X-ray or neutron diffraction, with respect to itsinability to identify the position of atoms in a unit cell, was completely eliminated by thedevelopment of CBED. The journey through the exciting evolution of this technique in thelater half of the previous century is very interesting.

The present paper provides a review of the fascinating developments in the area of CBED.The paper is formulated into the following sub-sections: (a) Historical perspective of thedevelopments in CBED, (b) basic principles, (c) additional information in CBED and (d)present status.

2. Historical perspective of the developments in CBED

Convergent beam electron diffraction is a technique with a long history of gradual develop-ment which has recently become widely available through the development of commercialTEM/STEM electron microscopes. Trials for detailed understanding of the concept of con-vergent beams were made by G Mollenstedt by building his own convergent beam electrondiffraction camera operating at a voltage of 45kV (Kossel & Mollenstedt 1939). It used awine bottle as the electron source. The vacuum at the specimen was 10−3 torr and the probesize was about 40µm. With such a large probe, contamination was not a problem despite apoor vacuum. Using flakes of mica as samples, he obtained very good CBED patterns.

For the first two decades, about from the 1940’s to the 60’s most of the efforts in TEMwere towards developing the capability of the microscopes by improving their design fea-tures. The progress in CBED development was rather slow and emphasis was mainly on thetheoretical development. MacGillavry (1940) used the two-beam theory to fit experimentalCBED patterns in the first attempt to measure structural factors using dynamical electrondiffraction theory. In the 1950’s, theoretical work on the dynamical theory was continuedby many researchers, and Kambe (1957) showed in his study on the three-beam theory thatintensity of diffracted beams depends on the sum of structural factors and could be measured.Throughout the sixties, the CBED method was developed by Lehmpfuhl in Berlin (Lehmp-fuhl & Reissland 1968), by Goodman (Goodman & Lehmpfuhl 1968) and by Moodie (1972)in Melbourne with the old, unsatisfactory microscopes. Gjonnes & Moodie (1965) explainedthe occurrence of forbidden reflections in the presence of strong multiple scattering, whichcould be used to identify the translational symmetry elements. During this period, Uyeda andHoier showed how the position of Kikuchi lines can be used to determine accelerating volt-ages and lattice constants (HOLZ lines in CBED patterns). At about the same time Uyeda(1968) and Watanabe (Watanabeet al1968) in Japan discovered the critical voltage effect* onKikuchi lines. Gjonnes & Hoier (1971) analysed this effect based on the three-beam theory

*This is the most accurate method for structure factor determination by the electron diffractiontechnique. Using an electron microscope with variable accelerating voltage, it is possible toobserve a minimum of intensity in the second-order reflection at the Bragg condition for aparticular voltage called the critical voltage, which is sensitive to the ratio of first- to second-orderstructure factors.

Convergent beam electron diffraction 765

and showed that the absence of intensity at certain points in these patterns may be used todetermine the three-phase invariant for centro-symmetric crystals. Thus it may be said thatmost of the important theoretical development of CBED was during the sixties.

By the early seventies, electron microscopes with very good design features like STEM,modified attachments, probes etc. had developed. An excellent review on the developmentsin electron microscopy is presented by Newbury & Williams (2000). During the same period,systematic procedures for determination of point and space group symmetry by CBED hadbegun to emerge due to the work by Goodman & Lehmpfuhl (1968), Steeds (1979) and Buxtonet al (1976). The theoretical foundations for point-group determination were established byBuxton and coworkers in the context of group theory who also developed the perturbationtheory for HOLZ interactions. In 1984, the fruitful result of the focused effort of Steeds groupin Bristol was the publication of an atlas of CBED patterns for various alloy phases (Steeds& Mansfield 1984). Starting from 1985, the high quality work of Tanaka’s group in Japanproduced excellent CBED patterns covering a wide range of applications and case studies(Tanaka & Terauchi 1985; Tanakaet al1988).

Ever since the applications of CBED have been growing. It is worthwhile recalling a few ofthem now. The Bristol group’s study on phase transformations in layer compounds supportingcharge-density waves brought the CBED technique to the attention of solid state physicistsfor the first time. The group was also successful in the structure determination of AuGeAsby CBED (Vincentet al 1984). At the same time, the technique was popularized in the USby the renowned work of Eades (1984) at Illinois. Throughout the late seventies and earlyeighties, Cowley inArizona was developing the coherent CBED method using sub-nanometreprobes to study regions of a crystal smaller than a unit cell (Cowley 1978a). Similar workon nanodiffraction, using imaging energy filters and novel detectors was later developed byBrown and co-workers in UK, which facilitated the study of defects in crystals (Brownet al1988). The use of CBED patterns to study line and planar defects also first began to bestudied at about this time. Large angle CBED (LACBED) for HOLZ and ZOLZ reflectionswere then developed in 1985 by Tafto and co-workers (Tafto & Metzger 1985). HOLZ effectsfrom artificial super lattices appeared first in the work of Chernset al (1988). The value ofshadow-imaging in CBED of HOLZ line intensities were realized during this period.

Research on structure factor phase measurements in non-centro symmetric crystals wasbegun in the mid-1980’s by Marthinsen, Hoier (1986, 1988) and others. Zooet al (1989)were successful in experimental structure–factor phase measurements, with accuracy betterthan one degree. At about the same time, measurements of local strains began to appear,reflected by the position of HOLZ lines with various dynamical correction schemes based onthe previous theoretical work by Jones and others (Jones & Hoier 1969) on dynamical shiftson Kikuchi lines.

Quantitative analysis of CBED patterns has become accurate after the use of elastic energyfiltering. This, together with the use of cooled CCD cameras, online work stations and figures,brings us to applying this technique to a wide range of problems in materials science, solidstate chemistry, mineralogy and condensed matter physics.

3. Basic principles

Conventionally, until the discovery of CBED, electron diffraction from a thin crystal in TEMwas obtained using a method called selected area diffraction-(SAD). The geometry of electrondiffraction is different in these two modes, namely SAD and CBED, as shown in figure 1.


(a) (b)

Figure 1. Electron diffraction in two different geometries.(a) Selected area diffraction (SAD) and(b) convergent beam electron diffraction (CBED).

It is seen that if a parallel beam of incident electrons is replaced by a convergent beam,diffraction spots of the SAD are enlarged into CBED discs. The same effect can also beintroduced by rocking the specimen. In the former, the incident electron beam is a parallelbeam of rays, with a single angle of incidence,α. In contrast, CBED makes use of a conicalbeam of electrons incident on the surface of the thin foil, with an angle of convergence asα.In such a case, the incident beam can be considered as a number of parallel incident beams,with a range of angles of incidence, from−αi to +αi . In SAD, the area from which thediffraction information is collected is selected by introducing a mechanical aperture in theimage plane. The demagnetized size of the aperture on the specimen plane defines the areafrom which diffraction information is collected. The smallest area from which diffractioninformation can be obtained using SAD is limited to 500nm due to the spherical aberrationof the objective lens and of the aperture (Leopold 1947). In CBED, the area for diffraction ischosen by focusing the incident beam into a very fine spot (2nm) on the region of interest.The angle of convergence is altered by changing the size of the condenser aperture. In boththe cases, the diffraction pattern is formed at the back focal plane (i.e.) of the objective lens,which is further magnified by a set of projector lenses. There is yet another mode of electrondiffraction called the ‘microdiffraction’, in which the angle of incidence is in between that ofSAD and CBED. The different modes of diffraction are identified by the relative value of theangle of convergence,α, to the Bragg angle of diffraction,θB , which is shown in figure 1. Itis also clear that the angle of convergence,α, is proportional to the diameter of the diffractiondisc in the diffraction pattern and the Bragg angle of diffractionθB , to the inter-spot/discdistance. These relations make it easy to calculate the angle of convergence experimentally.The comparison between the three modes of diffraction is summarized in table 1. The electrondiffraction patterns in the three modes are shown in figure 2. It is clearly seen that SADconsists of a set of spots, while microdiffraction gives a set of discs of small angular rangeand the CBED pattern consists of a set of discs with a higher angular range.

It is observed from the geometry of diffraction that SAD and CBED patterns are obtainedby interchanging the natures of the incident and the diffracted beams. That is, the incidentbeam in the case of SAD is a parallel beam, a disc of electron beams incident on the thin foiland all the diffracted beams are spots. In CBED, the incident beam is a spot on the surfaceof the thin foil, by virtue of the convergence introduced, and the diffracted beams are discs.Figure 3 shows the interchangeability of the incident and the diffracted beams in these twomodes of diffraction, the principle of which is referred to as reciprocity theorem.

The spatial resolution of CBED is limited not by the size of the incident beam but bythe broadening of the incident beam within the thin foil, especially in situations requiringdiffraction information from thick sections of the thin foil, when these effects are domi-nant. For all practical purposes, the resolution can be taken as a few tens of nanometres.


Table 1. Comparison between different geometries of diffraction.

Convergent beam electrondiffraction Microdiffraction Selected area diffraction

Small probe and large con-vergence

Small probe and very littleconvergence

Large probe size. No con-vergence

Large angular view of backfocal plane of objective. SoZOLZ, FOLZ and HOLZ canbe seen

Limited. Only ZOLZ andrarely FOLZ can be seen

Only ZOLZ

Finer details of intensity vs.angleθ due to crystal thick-ness and orientation differ-ences within diffracting vol-ume is not averaged

Intensity vs.θ is lost due toaveraging effects

Information lost

ZOLZ – zero order laue zone; FOLZ – first order laue zone; HOLZ – high order laue zone

When a gradual change in the geometry of diffraction is introduced from CBED to SAD,certain features emerge. The centres of the CBED discs coincide with the spot pattern, whenthe angle of convergence is decreased and a parallel incidence case is approached. It isknown that thickness and orientation are crucial in determining the intensity of diffractedelectrons.

Figure 2. Electron diffraction patterns along thezone axes indicated in the three modes: namely(a) SAD along [0 1 2] ofα-Al, (b) microdiffrac-tion along [0 0 0 1] of M2X in 9 Cr–1Mo steel and(c) CBED along [1 1 1] ofα-Fe.


Figure 3. Reciprocity theorem illustra-ting the interchangeability of the incidentand diffracted beams.

When the probe diameter is large, the variation in thickness and orientation within theilluminated area become so large that the useful information gets averaged out. This problemhas been overcome with the finer probes available. In CBED, the incident convergent beamcan be imagined as consisting of a large number of parallel beams. In such a case, for eachparallel beam, a diffraction spot is formed at the back focal plane of the objective lens, butaway from the optic axis. The distance of the diffracted spot from the optic axis dependson the inclination of the parallel beam under consideration. For every parallel beam, whichcontributes to the convergent beam, a diffraction spot is formed, depending on the angle ofincidence. Thus, the intensity distribution within the CBED disc provides information aboutthe angular dependence of diffracted intensity on the angle of incidence. Thus, the intensitydistribution within each of the CBED discs is a two-dimensional map of diffracted intensity asa function of the inclination between the incident electrons and a particular crystal direction.For every point in the (0 0 0) disc of the CBED pattern, there is a corresponding point in everyother diffracted disc, satisfying Bragg’s law.

4. Additional information in CBED

The phenomenal developments in the field of processing of metastable, novel microstructureshas offered challenging tasks in the unambiguous characterization of complex structures.Many available techniques need to be used in complementary ways to solve problems instructure analyses. Despite the development of techniques to identify structure, compositionand morphology simultaneously, the literature abounds with examples of cases, where uniqueidentification has been difficult. Distinguishing a spinodal product from a Guiner–Preston(GP) zone (Acuna & Bonfiglioli 1974), metastable phases like M23C6 and M6C in ferriticsteels (Tanakaet al1983a), voids and bubbles produced during irradiation (Brown & Mazey1964; Van Veenet al 1981), presence of low volume fractions of metastable fineδ-ferrite ina martensite matrix (Vijayalakshmiet al1999) are some typical examples.

CBED has been able to successfully solve some of the problems mentioned above. Thedistinction of M23C6 from M6C is illustrated here. Both the crystals have a face-centred cubicstructure with lattice parameters of 1·08 and 1·12nm respectively. These phases always formas minor constituents in steels (their amount< 0·1%), preventing the application of othertechniques like X-ray diffraction. The microchemistry of these carbides, even at equilibrium,can be quite varied depending on the chemistry of the steel. The lattice parameters are verysensitive to strain and composition. Though the equilibrium composition and lattice param-eters are well-documented, their validity for metastable phases is questionable. The onlydifference between the two phases, in all stages of their evolution, is the crystal symmetry:m3m for M23C6 andd3m for M6C. Determination of the symmetry by CBED thus is the


only confirmatory test to distinguish between the two carbides, which are often encounteredin steels. Another case where CBED alone is of help in distinguishing between two simi-lar phases is detecting the presence of fine, metastableδ-ferrite in a martensite matrix. Thismakes use of the measure of locked-in lattice strain in the two crystals, with much less strainin δ-ferrite compared to that in martensite.

Detailed analyses of CBED patterns could give good amount of additional and useful infor-mation about a crystal. These include (i) three-dimensional information about the reciprocallattice, (ii) point and space-group symmetry details, (iii) lattice parameters from regions asfine as 2nm, (iv) atom positions within a unit cell, and (v) defects in crystals. Each of theseis discussed in detail below.

4.1 Three-dimensional information about the reciprocal lattice

Generally, in electron diffraction, it is known that there are three equivalent methods torepresent the conditions for diffraction to take place: the Bragg’s law; Laue conditions andconstruction of the Ewald sphere (Kittel 1976). The last method is geometrical in nature.The reciprocal lattice points which lie on the surface of an imaginary sphere of radius 1/λ,(λ is the wavelength of the incident beam) satisfy the condition for constructive interferenceand intensity maximum occurs at these angles (figure 4). The shape of the diffraction peaksdepends on the size of the diffracting crystal and the angular width is inversely proportionalto the dimension of the diffracting crystal. Generally, the dimension of the thin foils used fortransmission electron microscopy along the ‘z’ direction is too small (thickness∼ a few nm).Hence, the angular widths of the diffraction peaks are large along the ‘z’ direction and show‘streaking’ along the direction of the incident beam. All the diffraction peaks obey the well-known ‘zone equation’ (figure 4) and the reciprocal layers are termed as either zero order lauezone (ZOLZ) or first order laue zone (FOLZ) and so on, depending on the value of ‘N ’ in thezone equation.

In conventional SAD mode, the angular view of the back focal plane (b.f.p.) of the objectivelens is generally confined to the ZOLZ. In the case of CBED, the angular view of the backfocal plane of the objective lens is enlarged significantly, due to the additional lens providedin the condenser-objective system. The difference between the two modes of diffraction, with

SPECIMEN

INCIDENT ELECTRONS

R = 1/λ

H

1/d

N = −1

N = 0 ZOLZ

N = −1 FOLZ

ZONE EQUATION: hu + kv + lw = ± N

Figure 4. Plan view of Ewald sphere construction forelectron diffraction.R represents the radius of the Ewaldsphere,λ is the wavelength of incident electrons,d is theinterlunar distance,H is the reciprocal lattice layer dis-tance, and ZOLZ, FOLZ, SOLZ refer to the zero, first andsecond order lauer zones respectively. The zone equationis the basis for this classification.


Figure 5. Sectional view of Ewald sphere showing radius of theFOLZ ring – G. H – reciprocal lattice layer distance, is related toGandλ.

respect to visibility of the b.f.p., of the objective lens, has already been illustrated in figure 2.The inter-zone distances are so small that their projection onto the observation plane canbe assumed to retain the geometrical relations betweenH, λ andG (figure 5). The planeof observation is usually the plane in which the diffraction pattern is recorded using thephotographic plate. A CBED pattern thus obtained, using a large angle of convergence froma large unit cell, consists of the following features: ZOLZ, whose discs are observed at thecentre, zero intensity for short angular distances, corresponding to the curvature of the Ewaldsphere between the ZOLZ and FOLZ, FOLZ which appears as a circle of discs which isactually the projection of the intersection of the Ewald sphere on the FOLZ. The same patternrepeats for SOLZ, HOLZ etc. A typical pattern is shown in figure 6. The inter-disc distancesin the ZOLZ when analysed exactly along the same lines as that of SAD provides informationregarding the two-dimensions of the reciprocal lattice. The corresponding two dimensions ofthe direct lattice can be derived from this information. Geometrical relations exist between theinterlayer distances, diameter of the FOLZ and the third dimension of the reciprocal latticeof simple crystals like face-centred cubic, body-centred cubic etc. Using these relations, thethird dimension can also be derived. Raghavanet al (1983), have used this principle to arriveat the complete description of the unit cell using a single CBED pattern.

4.2 Point and space-group symmetry details

The CBED pattern is a two-dimensional projection of the three-dimensional symmetry arounda crystal axis, along which the electron beam is incident. The CBED pattern symmetries can be

Figure 6. A typical CBED pattern along [0 66]ZA ofAl 6Mn showing HOLZ rings to derive three-dimensional information about the reciprocal lat-tice (ZA = zone axis).


described (Buxtonet al1976) by the 31 diffraction groups, which are isomorphic with Shub-nikov groups. The relations between the diffraction groups and the point groups of crystalsare also well-established. Hence, determination of the point group of a crystal, involves thefollowing steps: (i) identification of diffraction group of the crystal, along one close-packeddirection by study of the CBED pattern (figure 6) for whole-pattern symmetry, bright-fieldsymmetry and dark-field symmetry along+g and−g directions; (ii) repetition of the same,if required, along different zone axes, and (iii) derivation of the point group symmetry of thecrystal, based on diffraction symmetries. A number of references and illustrations are avail-able in the literature (Stoter 1981; Steeds & Vincent 1983a; Tanakaet al1983b) for the aboveapplication.

Space-group determination is carried out using the dynamical nature of electron diffrac-tion. When a crystal has a screw axis or glide planes, forbidden reflections occur near thekinematical condition of diffraction. These have finite intensities when dynamical conditionsof diffraction are operative. However, the cancellation of intensities, leading to extinction isstill caused for certain directions of the incident beam. Such an effect appears as dark lines inthe CBED discs that are called dynamic extinction lines or Gjonnes-Moodie (GM) lines. Thedynamic extinction effect is similar to the interference phenomenon in the Michelson inter-ferometer. That is, the incident beam is split into two beams by Bragg reflection in a crystal.These beams follow different paths, in which they suffer a relative phase shift when reflectedby crystal planes, and are then superposed on a kinematically forbidden reflection to canceleach other out. Detailed methods to determine space groups from these lines are discussedin many references (Steeds & Vincent 1983b; Tanakaet al 1983a). The details in the CBEDpattern are very sensitive to strain, defect and imaging conditions. Very often, these factorslead to error in judgment of the diffraction symmetries. Therefore, sufficient precaution mustbe taken while carrying out this exercise.

4.3 Determination of lattice parameter

The lattice parameter from microscopic regions is measured using CBED by making use ofcertain fine features of the pattern called the high order laue zone (HOLZ) lines (Joneset al1977). These lines appear as a pair called “deficiency lines” and “excess lines” (figure 7).The deficiency lines are seen in the (0 0 0) disc of the CBED pattern and the corresponding‘excess’ lines are seen in the discs of FOLZ. A pair of lines corresponds to a particular setof (h k l) planes of the crystal. The direction of these pair of lines is always parallel to eachother. The origin of HOLZ lines, procedure for imaging the HOLZ lines, precautions requiredwhile imaging, indexing of these lines and factors that govern the position of HOLZ lines arediscussed elsewhere (Vijayalakshmi 1997). The most important feature of relevance to thepresent discussion is that the angular position of these lines is sensitive to the acceleratingvoltage and the lattice parameter. Hence, if the accelerating voltage of the incident electrons ismaintained constant, the changes in the angular position of these HOLZ lines can be directlycorrelated to the variations in the lattice parameter.

CBED has been found to be the most appropriate technique in this laboratory for the studyof two problems, namely evaluation of lattice strain in microscopic regions (Vijayalakshmi1997; Saroja 1999) and the study of point defects in ion-irradiated crystals (Vijayalakshmi1997). A brief description of the first study is presented in the next section and the secondstudy later.

4.3aLattice strain measurements using CBED:The application of CBED for the identifica-tion and measurement of lattice strain in microscopic regions has been established (Spence


(c)

Figure 7. A typical CBED pattern ofα-Fe:(a) Whole pattern symmetry along[2 0 0] ZA, (b) HOLZ deficiency lines within (0 0 0) disc of (a), and(c) HOLZexcess lines in the discs of FOLZ ring of (a).

& Zuo 1992). The method has been standardised for 9Cr–1Mo steel in wrought and weldedconditions.

CBED patterns were obtained from the ferrite regions of 9Cr–1Mo steel with varyingdegrees of lattice strain due to their different thermal histories. In the case of weldment, thelattice strain variations with distance from the fusion zone were studied. The weldment wasdivided into four regions, the weld region, the heat-affected zone (HAZ) near the weld region,the HAZ close to the base metal and the base metal (figure 8). The percentage of change inthe uniform lattice strain was evaluated from the shift in the angular position of HOLZ linesin the CBED pattern (figure 9), taken along the same zone axes, from the four regions underidentical experimental conditions. Indexing of the HOLZ lines was done using EMS software.The distance between two chosen points on the HOLZ lines from the centre of (0 0 0) disc ismeasured and this distance is a measure of the change in the lattice parameter with respect toa reference (crystal with no strain). The detailed procedure for evaluation of lattice strain isgiven elsewhere (Vijayalakshmi 1997; Saroja 1999). The results obtained using CBED werecompared with the X-ray FWHM values.

WELD HAZ 1

HAZ 2

BASE METAL

Figure 8. Schematic representation of the regionsselected for study of variation of lattice strain byCBED experiments.


B

Figure 9. HOLZ deficiency lines in the (0 0 0) discof CBED pattern along [1 1 3] ZA ofα-ferrite in the(a) base metal (reference),(b) HAZ1 and(c) HAZ2.A andB are the points chosen for strain measurement.

4.4 Atom positions within a unit cell

For many decades, it was believed that electron diffraction is inferior to X-ray diffraction inproviding quantitative information on diffraction intensities (Cowley 1978b). This limitationof electron diffraction was found to be due to the strong interaction of electrons with thescattering elements, namely the electrons of the atoms in the diffracting crystal. This hasbeen termed the ‘dynamical diffraction’ effect. The consequence of the strong interactionof electrons with matter is that a one-to-one correlation could not be obtained between thescattering event and the intensity of diffracted beam, along a particular direction. As a result,no quantitative information could be obtained using electron diffraction, like the exact positioncoordinates of atoms within the unit cell of the crystal.

In order to obtain the above information, which is the ultimate in solving a crystal structure,intensity of diffraction under kinematical condition or weak scattering limits are required. Thecharacteristics of diffraction depend on the ratio of the values of the scattered and incidentenergies. If the scattered energy is very small relative to the incident energy, one can regardthe wave field after diffraction to be simply the addition of the incident, unperturbed wave andthe scattered radiation. This is called the ‘Born approximation’ or the ‘kinematic condition’of diffraction (Williams & Barry Carter 1996). X-rays and neutrons, by virtue of weak inter-action with matter, are kinematic in nature and therefore provide diffraction intensities underkinematic conditions. Hence, the procedure for determination of atom positions has beendeveloped in the early part of the last century, for X-ray and neutron diffraction. However, inthe case of electrons, the scattering is strong and the ‘Born approximation’ breaks down.

In the case of CBED, it is already shown that the larger visibility range of b.f.p. of theobjective lens provides information about intensity of diffraction about HOLZ discs. It is also


elasticI(ARB.UNITS)

θFigure 10. Angular distribution of the intensity ofelastically, scattered electrons.

known that the intensity of diffraction reduces as the scattering angle increases. Figure 10illustrates the angular dependence of scattered intensity vs. angle of scattering. Hence, theintensity of CBED discs at distances away from the optic axis or at very high angles ofscattering is kinematic in nature and can be used to extract information about the atompositions. The method developed using CBED for the determination of atom positions makesuse of this particular principle (Vincentet al1984).

Figure 7 shows the ‘excess lines’ in FOLZ discs. The intensity of many of these lines corre-sponding to many(h k l)’s is recorded and visually arranged as per their relative intensities ofdiffraction. Such information provides the database for the proposed exercise of identifyingthe atom positions. A detailed analysis of the origin of these ‘excess lines’ is carried out toidentify the ‘Bloch states’ that are responsible for their intensities and their relative strengthsof excitation. This information is translated into the required data by detailed computation,the details of which are given elsewhere (Ichimiya & Uyeda 1977).

4.5 Defects in a crystal

CBED has been extended for the analysis of planar defects like stacking faults, twin boundariesand grain boundaries. The detailed analysis of line defects like dislocations has also beensuccessful, using large angle CBED – LACBED (Tanakaet al1980; Tafto & Metzger 1985).The present section discusses briefly the signature in CBED patterns, due to the presence ofpoint defects, developed in the author’s laboratory.

CBED patterns from perfect crystal contain two fine features that show signatures of pointdefects. These features are: (i) the ‘interference pattern’ within the (0 0 0) disc of a perfectcrystal (figure 11), and (ii) the number of HOLZ rings (figure 6). It is to be kept in mind thatthe CBED pattern from a perfect crystal is a map of intensity of diffracted electrons. The three-dimensional crystal potential (figure 12a) can be considered as a projected two-dimensionalpotential (figure 12b) called the “string potential”. The projection scheme is valid since thetime available for the high energy, incident electrons, between the two parallel planes of atomsof the crystal, is very small. The strong interaction of the incident electrons with such a two-dimensional potential excites a number of plane waves called the “Bloch States” (figure 12c),the nature of which determines the intensity of diffraction. The ‘interference pattern’ withinthe (0 0 0) disc of a perfect crystal is caused by interference between the strongly excited“Bloch states”. The second signature, namely, the number of HOLZ rings in a perfect crystalis also very high, since the crystal is defect-free and the inelastic scattering due to defectsis practically minimum. The constituent atoms are in their equilibrium positions, leading tomaxima in the strengths of the diffracted peaks. Thus, the signal to noise ratio is high, causinga large number of HOLZ rings to be imaged.


Figure 11. CBED pattern ofα-ferrite along [1 1 3]ZA showing the interference pattern near the Bril-louin zone boundary (arrow marked).

When point defects are introduced in a perfect lattice, by deformation, quenching or irra-diation, the atoms are removed from their equilibrium positions and introduced in non-latticepositions. The lattice disorder reduces the strength of the periodic component of the scatter-ing potential and therefore reduces the intensity of Bragg diffraction. Similarly, there is anincrease in the diffuse background produced by elastic scattering from the aperiodic compo-nent of the crystal potential. The net result is to reduce the Bragg contrast, particularly forHOLZ reflections (figures 13a & b), which is associated with high-order Fourier coefficientsin the scattering potential. The effect of increasing the Debye–Waller factor is two-fold: first,the string potentials (projected atomic potentials) become less sharp (figure 14a & b) therebyreducing the amplitudes for the HOLZ diffraction. The string potentials also become some-what weaker, which induces some changes in the relative excitations and dispersions of theBloch states (figures 15a & b), which contribute to the contrast in lowest order reflections. Theoverall contrast is reduced because the CBED discs are viewed against the diffuse background.

Figure 12. Computed(a) three-dimensional crystal potential inα-Al along [1 1 2] ZA,(b) projectedtwo-dimensional potential inα-Al along [1 1 2] ZA and(c) Blochstates excited for an incident elec-tron of 100keV along [2 0 0] ZAof α-Al.


(a) (b)

Figure 13. Reduction in the number of HOLZ rings in CBED patterns ofAl6Mn due to ion irradiation(100keV argon ion):(a) unirradiated and(b) 2 × 1014 ions/cm2.

The above effects can also be simply viewed in terms of increase in the Debye–Waller(D–W) factor due to increase in ‘static mean square displacement’. The influence of increasein D–W factor as causing increase in inelastic scattering (noise) and reduction in elasticscattering signal is already very well-known. Hence, point defects increase the D–W factorand reduce the signal to noise ratio, leading to the observations stated above.

Thus, CBED patterns offer two signatures for the presence of point defects, which need tobe quantified for estimation of their concentration. The application of CBED for identifyingpoint defects is described below.

4.5aCBED for identification of point defects:The successful application of CBED for theidentification of point defects has been demonstrated for the first time in this laboratory.The systems chosen are the weld of 9Cr–1Mo steel and Al-14a/oMn alloy, consisting of twophases,α-Fe and M23C6 in the first case andα-Al and Al6Mn in the latter. Ion irradiation waschosen to introduce controlled amounts of point defects.

Two distinct features of CBED patterns that are sensitive to the concentration of pointdefects have been recognized.These are (i) the interference fringes in the (0 0 0) disk (figure 11)and (ii) the number of HOLZ rings (figure 6). These two features gradually reduce and disap-

Figure 14. Smearing of projected potentialalong [2 0 0] ZA inα-Al, with increase in DebyeWaller factor:(a) 0·005 and(b) 0·3 nm−1.


Figure 15. Smearing of Bloch states excited along [2 0 0] ZA inα-Al, for an incident electron of100keV, with increase in DW factor:(a) 0·005 and(b) 0·3 nm−1.

pear with increase in dose, show unique dependence on the mass and energy of incident ions,and reappear on post-irradiation annealing. In order to identify the fundamental cause of theabove observations, computations using EMS programs were carried out. These computationsshow that increase in the point defect concentration leads to smearing of projected potential,which in turn weakens and smears Bloch states that are excited. Consequently, the intensityoscillations of the interference pattern gradually disappear. The static displacement disorderreduces the strength of the large angle scattering, which is responsible for the reduction inthe number of HOLZ rings. Thus the present studies describes identification of two distinctfeatures sensitive to point defects and the understanding of the observed changes in terms ofthe projected potential, the excited Bloch states, the interference pattern in the CBED pattern,the static displacement disorder, the large angle scattering strength and the HOLZ rings.

4.6 Thickness determination

Sample thickness may be determined by a variety of methods in TEM like projected width ofinclined stacking faults and EELS spectra (Egerton 1986). The popular CBED method (Kellyet al1975) is based on the variation of the intensity of the diffracted beam with thickness knownas ‘Pendellosung’ fringes. Under a two-beam condition, the measurement of excitation errorat the positions of the fringe intensity minima can be used to determine the sample thickness,the method illustrated in several books on transmission electron microscopy (Spence & Zuo1992; Williams & Barry Carter 1996). This method has been widely used by many materialscientists today (Delilleet al2000; Bardalet al2000) as this offers accuracy better than 2%.

5. Present status of CBED

In real materials defects, precipitates and local strains occur on a very fine scale so it becamevery essential to develop methods for obtaining smaller electron probes to minimize thecontribution from defects, where conventional CBED fails. (Cowley 1978, 1992; Spence &Carpenter 1986; Brownet al 1988). The development of FEG (Tsong 1990) has contributedgreatly to this. CBED patterns show high contrast as the probe size is reduced and furtherenhancement in contrast is obtained by using an energy filter tuned to the elastic peak in theenergy loss spectrum (Lanio 1986). This could lead to improvements in critical-voltage mea-surements, lattice-parameter determination and in the resolution of branch cluster information(Midgley et al 1995; Kramer 2000). As the probe size becomes smaller, the electron beam


becomes fully coherent (Spence & Zuo 1992), which has led to the development of a varietyof techniques for various problems in materials science. A few of them are discussed here.

5.1 Nanodiffraction

This is a special form of CBED, in which the emphasis is on obtaining diffraction patternsfrom regions of the specimen, about 1nm or less in diameter. Unless a field-emission gun(FEG) is used, the intensity in a beam 1nm in diameter is too small to be useful. Hence,nanodiffraction has been performed mainly in dedicated STEM instruments having coldFEG sources, although the newer TEM’s with FEG’s may also be used with efficient two-dimensional detector systems and recording with TV or CCD cameras (Cowley 1991). In thecase of conventional CBED, the probe is incoherent and comparatively large. However, innanodiffraction the electron beam is perfectly coherent with diameters as small as 0·2nm,because of which crystallographic information on a near atomic scale can be obtained, witha wide range of applications. These include structure analysis of metal particles in catalysts(Cowley & Plano 1987; Panet al1987), study of defects (twins, dislocations etc.) and disorderin very small particles (Monosmith & Cowley 1984), use of HOLZ and Kikuchi line splittingto determine fault vector of the defect (Gjonnes 1985), structure of individual defects inthin crystal foils (Cowleyet al 1984), determination of the local order in thin films of nearamorphous materials or disordered crystals (Chan & Cowley 1981; Ohkuboet al 2000) anddetermination of local symmetry within particular parts of a unit cell of a crystal or a defect.A mixed mode operation of the microscope as in the convergent beam imaging (CBIM) andLACBED methods which produce a shadow image of the sample superimposed on the CBEDpattern can also be used for the study of defects (Spence & Zuo 1992).

5.2 Lattice imaging

The discs in CBED patterns, formed using coherent nano probes with a large convergenceangle are allowed to overlap, and the overlapping region of the pattern reveals the latticefringes, which leads to STEM lattice imaging (Spence & Cowley 1978) (figure 16). It ispossible to locate the probe accurately by this method at various regions within the unit celland the CBED patterns obtained from these areas show different site symmetries (Ouet al1989) and atomic positions. A coherent CBED pattern recorded with a very large objectiveaperture or without the objective aperture, so that a gross overlap of CBED discs occurs iscalled a ‘ronchigram’. In fact, as an extreme case, if an ideal point source was available, anarrangement in the form of an ‘X’ would produce an unaberrated lattice image of the crystalwithout using either lens or scanning.This point projection method for electron lattice imagingwas proposed by Cowley & Moodie (1957). These resulting images are called Fourier imageswhich are now being obtained even at very low accelerating voltages (∼ 300 volts) using asputtered tungsten field emission tip instead of a focused probe as a point electron emitter,without any lenses (Finket al1991).

5.3 Electron holography

Though invented in 1948 by Gabor, it came into prominence only in the 1990’s. An electronhologram of an object is the interference pattern of the elastically scattered (diffracted) wavefrom the object and a reference wave, both resulting from a single primary wave and therebyachieving synchronism. The key feature of an electron hologram is that, unlike conventionalTEM, both phase and amplitude of the beam can be recorded which enables us to studymany phase dependent phenomena like magnetism (Tonomura 1987; Volkov & Zhu 2000;


Figure 16. Axial three-beam lattice imaging in STEM, with an illumination angle twice the Braggangle. Three orders overlap atD. The appearance of a two-dimensional coherent CBED pattern usedfor axial five beam imaging is shown at the right.

Shindoet al2002) with a very high resolution. Several electron optical geometries have beendeveloped for making electron holograms (Cowley 1992), but the most popular one is the“off-axis, image plane” geometry. It employs the electron biprism, invented by M¨ollenstedtand Duker in 1955. This device is simply an ultra-fine (0·3µm in diameter) conductive fibrepositioned in an imaging lens perpendicular to the electron beam so that it splits the field ofview.A thin TEM specimen is placed over one side of the image field. When a positive voltageis applied to the fibre, the electron waves on either side of the fibre are bent toward the centre,eventually causing them to overlap. The overlapping waves create an interference pattern ofparallel fringes. These fringes are changed in position and contrast, depending upon how thespecimen affects the electron beam. The pattern is recorded either on film, or directly on toa digital CCD camera system. This interferogram, or hologram, is then processed to yieldseparate amplitude and phase images to give an atomic resolution (Orchowskiet al1995).

The coherent nano diffraction from thin crystals called the Gabor inline holography needs apoint source and a weakly scattering transmission object.A number of reconstruction schemes


using coherent microdiffraction patterns are being attempted (Lehmann & Lichte 1995) Todayelectron holography has become a necessary tool in the study of nanocrystalline and magneticmaterials (Beeliet al 1999), interfaces (Weisset al 1993), heterostructures (Rosenaueret al2001) and quantum wells (Chernset al1999).

6. Conclusions

The present paper is an overview on the technique of CBED summarizing various featureslike its discovery, historical development, principle, information obtained and its presentstatus. The study of very small particles that could be as small as a single unit cell but couldinfluence the mechanical and electrical properties of many technologically important materi-als like catalysts, microphases at interfaces etc. is not possible without a fine probe. Electronmicroscope instrumentation has undergone continuous developments for study of a widerange of materials problems. Today it is possible to observe directly the shape of electronorbitals and thus the shape of molecules. This is a very important step towards understandingchemical bonding, which holds matter together. Such information is of utmost significancenot only for chemical science but also for applications related to materials properties, forexample in the field of superconductors, semiconductors, nanomaterials, ceramics etc. Hence,it can be said with confidence that these advanced techniques and their development willpave the path for structure analysis problems for newer materials in future.

The authors wish to acknowledge the constant support and encouragement of Dr V S Raghu-nathan and Dr Baldev Raj.

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